{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $S$ be a smooth projective [[algebraic surface|KFTECAATKG]] over $\\C$ that is minimal, meaning that it contains no smooth rational curve $E$ with $E^2 = -1$.\nLet $\\kappa(S)$ be its [[Kodaira dimension|ZW00Q0SAGR]] and $K_S$ its [[canonical divisor|W05VYY4HBR]].\nThen exactly one of the following holds.\n\n1. $\\kappa(S) = -\\infty$, and $S$ is the projective plane $\\mathbb{P}^2$ or a [[ruled surface|5NK10NGY13]] $\\mathbb{P}(\\mathcal{E}) \\to C$ over a smooth projective curve $C$.\n2. $\\kappa(S) = 0$, and $S$ is a [[K3 surface|Q3V566R6K2]], an [[Enriques surface|Y30JE2DC6Y]], an [[abelian surface|R7W03TXRS9]] or a bielliptic surface. In each case $12 K_S \\sim 0$.\n3. $\\kappa(S) = 1$, and $S$ is an [[elliptic surface|VSC3F53DX4]].\n4. $\\kappa(S) = 2$, and $S$ is a [[surface of general type|SV4CK2XP8M]].\n\nMoreover, every smooth projective surface is obtained from a minimal one by finitely many blow-ups of points, and if $\\kappa(S) \\ge 0$ this minimal model is unique.\n\n## Remarks\n\nThe reduction to minimal surfaces uses [[Castelnuovo's contraction theorem|WJ5WRGG5DE]]: a smooth rational curve with self-intersection $-1$ can be blown down to a smooth point, which lowers the second Betti number.\nFor each value of $\\kappa$, the finer classification uses the invariants $q = h^1(\\mathcal{O}_S)$, $p_g = h^0(K_S)$, the plurigenera $P_n = h^0(nK_S)$, and the Chern numbers $K_S^2$ and $c_2(S)$.\n[[Castelnuovo's rationality criterion|QXHMMQX1D7]] is the key step for $\\kappa = -\\infty$.\nKodaira extended the classification to all compact complex surfaces, which adds non-algebraic classes such as surfaces of class VII and Kodaira surfaces.\nBombieri and Mumford extended it to algebraically closed fields of positive characteristic, where new classes appear in characteristics $2$ and $3$.\nExample: a smooth quartic surface in $\\mathbb{P}^3$ has $K_S = 0$ by adjunction and lands in case 2 as a K3 surface.\nA smooth quintic surface has $K_S$ ample and lands in case 4.\n\n## Explanations\n\n### L1 human\n\nEvery complex algebraic surface can be simplified, by collapsing certain special curves, to a minimal surface.\nThe classification sorts minimal surfaces into four families according to how fast the number of sections of powers of the canonical bundle grows.\nThe first three families are known in detail; the last one, the surfaces of general type, is by far the largest and is not classified in detail.\n\n### L2 human\n\nThe classification is the two-dimensional model of the minimal model program.\nIn dimension one, curves split by genus into $g = 0$, $g = 1$ and $g \\ge 2$, which matches $\\kappa = -\\infty, 0, 1$; for surfaces there is one more class.\nThe invariants that tell the classes apart are birational invariants, so they can be computed on any smooth model.\nResults like the Bogomolov–Miyaoka–Yau inequality constrain the invariants of the last class.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "KFTECAATKG",
      "ZW00Q0SAGR"
    ],
    "kind": "statement",
    "msc": [
      "14J10",
      "32J15"
    ],
    "name": "Enriques–Kodaira classification",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q2584390",
      "wikipedia": "Enriques–Kodaira classification"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Enriques%E2%80%93Kodaira_classification"
      }
    ],
    "tag": "1GVF19KFAK",
    "uses": [
      "KFTECAATKG",
      "ZW00Q0SAGR",
      "W05VYY4HBR",
      "5NK10NGY13",
      "Q3V566R6K2",
      "Y30JE2DC6Y",
      "VSC3F53DX4",
      "SV4CK2XP8M",
      "R7W03TXRS9",
      "WJ5WRGG5DE",
      "QXHMMQX1D7"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/1GVF19KFAK/",
  "links": [
    {
      "field": "about",
      "target": "KFTECAATKG"
    },
    {
      "field": "about",
      "target": "ZW00Q0SAGR"
    },
    {
      "field": "uses",
      "target": "KFTECAATKG"
    },
    {
      "field": "uses",
      "target": "ZW00Q0SAGR"
    },
    {
      "field": "uses",
      "target": "W05VYY4HBR"
    },
    {
      "field": "uses",
      "target": "5NK10NGY13"
    },
    {
      "field": "uses",
      "target": "Q3V566R6K2"
    },
    {
      "field": "uses",
      "target": "Y30JE2DC6Y"
    },
    {
      "field": "uses",
      "target": "VSC3F53DX4"
    },
    {
      "field": "uses",
      "target": "SV4CK2XP8M"
    },
    {
      "field": "uses",
      "target": "R7W03TXRS9"
    },
    {
      "field": "uses",
      "target": "WJ5WRGG5DE"
    },
    {
      "field": "uses",
      "target": "QXHMMQX1D7"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/1GVF19KFAK.md",
  "msc": [
    "14J10",
    "32J15"
  ],
  "name": "Enriques–Kodaira classification",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1GV/1GVF19KFAK.md",
  "status": null,
  "tag": "1GVF19KFAK",
  "uri": "m2://1GVF19KFAK",
  "url": "https://tamnd.github.io/math2/wiki/enriques-kodaira-classification/",
  "writing": {
    "tag": "1GVF19KFAK",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}